Principles of Cosmic Particle Acceleration · Acceleration in the Sun - moving magnetic ﬂuxes...

Introduction Acceleration Limits Sites End

Principles of Cosmic Particle AccelerationSeminar Nuclear Astrophysics

”Nuclei in the Cosmos - Accelerators throughout the Universe”

Herbert Kaiser

November 28, 2007


Outline

1 IntroductionIntroduction

2 Acceleration MechanismsChanging Magnetic FieldsMirror Effect2nd Order Fermi AccelerationDiffusive Shock Acceleration

3 LimitsAcceleretion Limits

4 SitesAcceleration Sites


Introduction

Accelerated Particles - Cosmic Rays

high energy charged particles reaching the Earth’s atmosphere:

electrons ∼ 1%

protons ∼ 89%

heavier nuclei, mainly helium ∼ 10%

very few: antiparticles, muons, pions, kaons (from interactionsof CRs with the interstellar gas)


Introduction

CR Spectrum I

power law:N(E) ∝ E−γ


Introduction

CR Spectrum II

knee:particles withE > 1015 eV start toleak from the galaxy,maximum fromsupernova explosionsnear 1015 eV

ankle:extragalacticparticles

GZK-cut off at≈ 6 · 1019 GeV


Introduction

Isotropy - Anisotropy

charged particles ⇒ circular orbits in a magnetic field

gyro-radius:

rg =p

qB=

γm0 v

ZeB

CR sky for energies < 1014 eV is completely isotropic

high energy CRs are not as much deflectedexample: E = 1019 eV, B = 2µG, rg = 10 kPc

⇒ anisotropy for high energies?


Introduction

Possible Cosmic Ray Sources - Magnetic Fields

Hillas 1984

measurement:polarizationchanges of radiowaves due toFaraday effect


Introduction

Newest Auger Collaboration Results

circles - one detetected particle >= 56EeV, stars - possible sources(AGNs, distance < 75 MPc), dashed line - supergalactic plane,solid line - border of the field of view


Outline






Changing Magnetic Fields

Cyclotron Mechanism

Faraday’s law

∇× E = −B

decaying magnetic field ⇒ electric field ⇒ acceleration

magnetic flux: Φ = B ·A = BπR2

change of the flux ⇒ potential:

U =

∮E · ds = −Φ = −BπR2

energy gain

∆E = eU = eπR2B

from one gyration



Sunspot



Acceleration in the Sun - moving magnetic fluxes

magnetic flux tubes reaching the surface

particles cannot move perpendicular to magnetic field lines

inhibited energy flux ⇒ lower temperature ⇒ darker ⇒ sunspot

occurence as moving pairs

eg.: merging in order to reduce the magnetic field energy

energy estimate:

flux tube radius R ≈ 104km

magnetic field B ≈ 2000 G

merging in 1 day,B = 2000G/day

⇒ E = 0.73 GeV



Sunspot Loops



Magnetic Reconnection

approaching/merging of magnetic flux tubes

magnetic pressure low in the middle ⇒ plasma inflow

magnetic field lines “like” to be straight⇒ acceleration to the left and to the right

magnetic field energy converted to plasma velocity



CRs from the Sun

measurements duringvarious levels of solaractivity

maximum around 0.5 GeV

power law at high energies


Mirror Effect

Mirror Effect - Overview

particles entering regions of higher magnetic field strength arereflected backwards

charged particles follow cyclotron orbitsgyro-radius: rg = p⊥/qB

Rule

The magnetic fluxΦ = B · πr2

g ∝ v2⊥/B

through the particles’cyclotron circle isconstant.

stronger magnetic field⇒ smaller gyro-radius, increased perpendicular velocity v⊥⇒ decrease of parallel velocity v‖ (energy conservation)⇒ v‖ → 0, then reflection


Mirror Effect

Mirror Effect - Reflecting Force

charged particle on a circular orbit⇒ magnetic dipole moment ~µ normal to the circle

energy of a magnetic dipole: E = −~µ · ~B⇒ favourable in terms of energy: ~µ ‖ ~B⇒ position near a B-gradient always unfavourable⇒ reflecting force


Mirror Effect

Magnetosphere

radio wave echos were used to detect charged particles trapped inthe magnetosphere


Mirror Effect

Reflection and Acceleration

reflection on a moving object with velocity u

change in kinetic energy:∆E = 1

2m(v + u)2 − 12mv2

relativistic calculation →

∆E ∝ E,∆E

E= 2

uv

c2+

u2

c2

v≈c= 2

u

c︸︷︷︸1st order(±)

+ 2u2

c2︸︷︷︸2nd order

1st order term → 1st order Fermi mechanism:reflections/scattering in a shock front

2nd order term →2nd order Fermi mechanism :statistical reflection on many different clouds in a galaxy

scattering by magnetic irregularities: plasma waves (Alfven waves..)


2nd Order Fermi Acceleration


acceleration by stochastic reflection on moving objects

Fermi 1949: acceleration by reflections on moving magneticfields in the galaxy

probability for a head on collision is larger than for anovertaking collision

1st order term in ∆E/E cancels out

energy gain

average over all angles of incidence ϑ:⟨∆E

E

⟩∝ u2

v2



2nd Order Fermi Acceleration - Energy Spectrum

particles will be accelerated by reflections in the galaxy as long asthey don’t lose more energy on the way than gaining by thereflections

energy increases exponentially with # of reflections (∆E ∝ E)

mean acceleration time T before a catastrophic particleinteraction⇒ “age” distribution: f(t)dt = 1

T e−t/T dt

average time between two reflections τ

Result: power law

f(E) ∝ E−γ , γ = 1 + τ/

(u2

c2T

)important: T needs to be large enough ⇒ particles injected to

the mechanism need sufficiently high energies



2nd Order Fermi Acceleration - Fermi’s estimates

absorption mean free path for 200 MeV in interstaller matter isl = 107 Ly⇒ long enough for multiple collisions

average acceleration time (for v = c) is T = l/t = 6 · 107 y

experiments (at Fermi’s time) gave:

γ = 2.9 = 1 + τ/

(u2

c2

)T

⇒ τ = 1.3 y average time between two reflections


Diffusive Shock Acceleration

Diffusive Shock Acceleration - Conditions

a shock front moving with velocity u

magnetic fields near the shock front→ particle confinement (see mirror confinement)

plasma waves (Alfven..) → scattering → diffusion

no dense gas needed (e.g. 1 particle/cm3):

particles interact by electromagnetic fields in the plasma

small em. field fluctuations in the plasma will grow in the shock

particles at higher energy (supra-thermal or injected particles)



Diffusive Shock Acceleration - Principle

picture in the rest frame of the shock front

particles with higher velocity than the plasma flow may travel against the stream

reflection in upstream ⇒ energy gain ∝ vup/c

reflection in downstream ⇒ smaller energy loss ∝ vdown/c

repetition until particle is not scattered back upstream

loss of particles downstream in each cycle: ∆Nloss/N ∝ u/c



Diffusive Shock Acceleration - Features

∆E/E 1st order in u/c, Fermi I more efficient than Fermi II

downstream loss ∆Nloss/N =const.⇒

energy increases exponentially with # of cycles

# of participating particles N decreases exponentially with # ofcycles

power law for E > Einj

f(E) ∝(

E

Einj

)−γ(M)

θ(E − Einj)



Diffusive Shock Acceleration - Spectrum

mach number M = vupstream/csound


Outline






Acceleretion Limits

Limits to Acceleration

increasing E ⇒ increasing gyro-radius⇒ limit to the maximum energy (see Hillas diagram)

energy gains have to exceed the losses:radiative: synchrotron radiation, bremsstrahlung, inverse

Compton scattering

non-radiative: coulomb scattering, ionization

catastrophic: hadronic interactions: p + p. . .

GZK-cut-off: interaction with CMB photons

most of the electrons will immediately lose their energy byradiation losses in the em. fields in the shock⇒ synchrotron radiation = hint to acceleration site


Acceleretion Limits

Possible Cosmic Ray Sources - Magnetic Fields

Hillas 1984

measurement:polarizationchanges of radiowaves due toFaraday effect


Acceleretion Limits

Limits to Acceleration

increasing E ⇒ increasing gyro-radius⇒ limit to the maximum energy (see Hillas diagram)

energy gains have to exceed the losses:radiative: synchrotron radiation, bremsstrahlung, inverse

Compton scattering

non-radiative: coulomb scattering, ionization

catastrophic: hadronic interactions: p + p. . .

GZK-cut-off: interaction with CMB photons

most of the electrons will immediately lose their energy byradiation losses in the em. fields in the shock⇒ synchrotron radiation = hint to acceleration site


Outline






Acceleration Sites

Acceleration Sites

diffusive shock acceleration:

structure formation shocks (eg. in merging galaxy clusters)

supernovae

shock fronts in active galagtic nuclei (AGN), near black holes orneutron stars

galactic winds

turbulences in the sun

2nd order Fermi acceleration:

near a shock front (smaller contribution u2

c2 uc )

scattering at magnetic irregularities and plasma waves

confinement at shock fronts

clouds in a galaxy, solar winds. . .


Acceleration Sites

Structure Formation Shocks

Galaxy cluster Abell 3376

Mpc-scale supersonic radio-emitting shockwaves

radio sources (synchrotron radiation..) may be acceleration sitesboosting particles up to 1019 eV

hints to subcluster merger activities


Acceleration Sites

Supernova Remnant (SNR)

Cassiopeia A (exploded 300 years ago)

x-ray picture - hot gas radio picture - synchrotron radiation


Conclusion - some keywords

CR spectrum, power law, anisotropy?

presented acceleration mechanisms:

changing magnetic fields, magnetic reconnection

mirror effect + reflection on moving objects

Fermi II

Fermi I - diffusive shock acceleration

limits: gyro-radius, energy losses

acceleration sites: structure formation shocks, SNR, etc.


References

Grupen, C. Astroparticle Physics, Springer 2005

Schlickeiser, R. Cosmic Ray Astrophysics, Springer 2002

Grimani, C. et al. Cosmic-ray spectra near the LISA orbit, IoP,Class. Quantum Grav. 21 (2004) S629

Fermi, E. On the Origin of Cosmic Radiation, Phys. Rev. Vol.75, 8 (1949) 1169

Pfrommer, C. On the role of cosmic rays in clusters ofgalaxies, PhD-thesis 2005, LMU Munchen

NASA, NASA image gallery,http://www.nasa.gov/multimedia/imagegallery/index.html

Bagchi, J. et al., Abell 3376. . . ,http://www.sciencemag.org/cgi/content/full/314/5800/791


Mirror effect - Systematic Treatment

Use adiabatic invariants: J =∮

P · dQ

generalized momentum P = mv + qA, B = ∇× A

with B = B ez, we have A = 12Br eϕ

particle moves along an orbit with Q = r = rg = v⊥/Ω

J =

∮ (mv⊥ −

1

2qBrg

)rgdϕ = 2π

(mΩr2

g −1

2mΩr2

g

)= q πB r2

g︸︷︷︸=Φ

=πm2

q

v2⊥B

v2⊥/B is constant while the particle is moving to larger B⇒ v⊥ must increase and because of the conservation ofkinetic energy v‖ must decrease


Mirror effect - Constraints for Reflection

certain amount of v‖ is converted to v⊥, v2⊥/B is constant

⇒ difference Bm −B0 has to be large enough for v‖ → 0⇒ v⊥/v‖ has to be large enough (v‖ not too large)

pitch angle ϑ = arctanv⊥v‖

!> arctan

√B0

Bm −B0

similar to reflection on a bottle neck


Energy Changes from Reflections I

relativistic treatment of a particle reflection (2 dimensions: t, x)

in the observer’s frame the momentum is

p(µ) =

(E/cp

)= γvm0

(cv

)in the gas cloud’s rest frame we have

p′(µ) = Λu p(µ) = γvγum0

(1 u/c

u/c 1

)(cv

)= m0γvγu

(c + uv

cu + v

)after reflection p

′(µ)rf = m0γvγu

(c + uv

c−u− v

)again in the observer’s frame

p(µ)rf = Λ−u p

′(µ)rf = m0γvγ

2u

(c + 2uv

c + u2

c

−v − 2u− u2vc2

)


Energy Changes from Reflections II

energy ratio is: ErfE =

1+2uvc2 +u2

c2

1−u2

c2

≈ 1 + 2uvc2 + 2u2

c2 (for u c)

⇒ relative energy change is:

∆E

E= 2

uv

c2+

u2

c2

v≈c= 2

u

c︸︷︷︸1st order(±)

+ 2u2

c2︸︷︷︸2nd order

, ∆E ∝ E

1st order term → 1st order Fermi mechanism:

reflection/scattering by a shock front

2nd order term →2nd order Fermi mechanism :

statistical reflection on many different clouds

scattering by magnetic irregularities: plasma waves (mostlyAlfven waves)


Diffusive Shock Acceleration - Transport Equation

Fokker-Planck type equation for the distribution f(t, x, p):

∂

∂tf+

∂

∂xv(x, p) f = − 1

p2A(x, p) f+

1

p2Γ(x, p)

∂

∂pf+

∂

∂xD(x, p)

∂

∂xf+s(x, p)

D - diffusion coefficient: 〈δx2〉 = 2D∆t∆x = v∆t + δx, v= medium bulk motion

A - describes energy losses/gains by 1st order processes in v/c,examples:

Aradiation = 〈∆p〉∆t |rad ∝ 〈me

m 〉2γ2

compressed flow (∇ · v < 0) → acceleration force Fj = −pi∂vj

∂xi

Eacc = −pv3 ∇ · v ⇒ Aacc = −p

3∂v∂x

Γ - energy gain through 2nd order Fermi: Γ = 〈δp2〉2∆t ∼

v2A

c2 νsp2

s - source term (injected CRs)


Diffusive Shock Acceleration - Stationary Solution

starting with a narrow CR injection distribution

g(p) =ηCR n

4πp2inj

δ(p− pinj)

we get a CR population that again obeys a power law:

f(p) =α ηCR n

4πp3inj

(p

pinj

)−α

θ(p− pinj)

ηCR - injection efficiency (∼ 10−4)

α = 3r/(r − 1), r =γ + 1

γ − 1 + 2/M2

γ - adiabatic index γ = 5/3 or γ = 4/3 (relativistic)

M - Mach number M = vupstream/csound


Radiation and inverse Compton losses

CR + γ → CR + γ′

inverse Compton scattering:relativistic charged particles are scattered by photons

lose part of their kinetic energy to the photon

synchrotron losses:charged particles in a magnetic field are forced on circular orbits

deceleration/acceleration ⇒ emission of photons

bremsstrahlung:deceleration in a Coulomb field

important for electrons, for other particles suppressed by(me/m)2

proportional to γ2 ⇒ more important at high energies


Coulomb and ionization losses

scattering in the Coulomb field of particles in the gas

scattering by bound electrons →ionization

most important losses for protons and heavier nuclei,less important for electrons

more mass ⇒ more losses (cf. Bethe-Bloch formula)

higher energies ⇒ smaller cross section ⇒ fewer losses

⇒ for stochastic acceleration processes (Fermi II) high injectionenergies are needed in order to exceed the losses

proton: ≈ 200 Mevoxygen: ≈ 20 Gev


Catastrophic losses, GZK cut-off

CRs with kinetic energy beyond 0.78 GeV can interact hadronically

CRs above an energy of 6 · 1019 GeV interact with CMB photons:

γ + p → p + π0, γ + p → n + π+

(in the proton rest frame CMB photons have high enough energies)

mean free path for protons beyond that energy is about 10 Mpc

GZK cut-off is higher for heavier nuclei


Detection of Acceleration Sites

Principles of Cosmic Particle Acceleration · Acceleration in the Sun - moving magnetic ﬂuxes...

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